$4$

ROC Curve Analysis $(10$ points$)$

A company uses a security device that has a characteristic $($or ROC curve given by

Pa $=0\mathrm{.}7($Ps$)0\mathrm{.}5$

Assume that the cost of a misdetection is $$10,$ while that of a false positive is $$2.$ The base rate of attacks is $0\mathrm{.}05.$ What is the optimal operating point for the device?